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<meta name="description" content="大家还好吗？背景就不用多说了吧？本来我是初四上班的，现在延长到2月10日了。这是我工作以来时间最长的一个假期了。可惜哪也去不了。待在家里，没啥事，就用python模拟预测一下新冠病毒肺炎的数据吧。要声明的是本文纯属个人自娱自乐，不代表真实情况。采用SIR模型，S代表易感者，I表示感染者，R表示恢复者。染病人群为传染源，通过一定几率把传染病传给易感人群，ta自己也有一定的几率被治愈并免疫，或死亡。易">
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          <h1 class="post-title" itemprop="name headline">SIR模型预测新冠病毒肺炎发病数据</h1>
        

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        <p>大家还好吗？<br>背景就不用多说了吧？本来我是初四上班的，现在延长到2月10日了。这是我工作以来时间最长的一个假期了。可惜哪也去不了。待在家里，没啥事，就用python模拟预测一下新冠病毒肺炎的数据吧。要声明的是本文纯属个人自娱自乐，不代表真实情况。<br>采用SIR模型，S代表易感者，I表示感染者，R表示恢复者。染病人群为传染源，通过一定几率把传染病传给易感人群，ta自己也有一定的几率被治愈并免疫，或死亡。易感人群一旦感染即成为新的传染源。<br>模型假设:<br>①不考虑人口出生、死亡、流动等情况，即人口数量保持常数。<br>②一个病人一旦与易感者接触就必然具有一定的传染力。假设 t 时刻单位时间内，一个病人能传染的易感者数目与此环境内易感者总数s(t)成正比，比例系数为β，从而在t时刻单位时间内被所有病人传染的人数为βs(t)i(t)。<br>③ t 时刻，单位时间内从染病者中移出的人数与病人数量成正比，比例系数为γ，单位时间内移出者的数量为γi(t)。<br>模型为<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/01.png"><br>其中，β为感染系数，代表易感人群与传染源接触被感染的概率。γ为隔离(恢复)系数，我们对其倒数1/γ更感兴趣，代表了平均感染时间(average infectious period)。S(0)为初始易感人数，I(0)为初始感染人数。<br>按照[1]里面的代码模型的感染人数是这样的<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/02.png"><br>现在的问题就是利用现有的数据找到新冠肺炎的β值，γ值等数据了。先把数据拔下来吧。从[3]上扒数据，由于数据不多，就手工完成吧。保存到csv文件里。<br>然后把数据作图<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/03.png"><br>还有一个指标是再生数R0=β/γ，大于1时人群中大部分才被感染[4]。世卫组织1月23日的估计是R0在1.4到2.5之间[5]，最新的根据前425例发病数据的估计值为2.2[6]。<br>文章[7]中的按一般病毒性肺炎恢复期25天计算得到的γ值为0.04。<br>关于β值和初始易感人群，[7]的作者采用的方法是先估计一个区间，然后用最小二乘法找到最佳参数，β≈3.57*10^-5。S[0]的范围为5000-30000人。[7]文章里有matlab代码，我用python改写一下，由于对最小二乘法法的实现比较陌生，尝试了半天，最后我决定用最笨的办法——穷举法。就是用两个嵌套循环将范围内所有β值和S0值都试一遍，计算每次尝试结果与实际数据之间差值的平方和，平方和最小的一组β值和S0值用来做预测。代码如下:</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># γ值设定为0.04，即一般病程25天</span></span><br><span class="line"><span class="comment"># 用最小二乘法估计β值和初始易感人数</span></span><br><span class="line">    gamma = <span class="number">0.04</span></span><br><span class="line">    S0 = [i <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(<span class="number">20000</span>, <span class="number">40000</span>, <span class="number">1000</span>)]</span><br><span class="line">    beta = [f <span class="keyword">for</span> f <span class="keyword">in</span> np.arange(<span class="number">1e-7</span>, <span class="number">1e-4</span>, <span class="number">1e-7</span>)]</span><br><span class="line">    <span class="comment"># 定义偏差函数</span></span><br><span class="line">    <span class="function"><span class="keyword">def</span> <span class="title">error</span>(<span class="params">res</span>):</span></span><br><span class="line">        err = (data[<span class="string">&quot;感染者&quot;</span>] - res)**<span class="number">2</span></span><br><span class="line">        errsum = <span class="built_in">sum</span>(err)</span><br><span class="line">        <span class="keyword">return</span> errsum</span><br><span class="line"></span><br><span class="line">    <span class="comment"># 穷举法，找出与实际数据差的平方和最小的S0和beta值</span></span><br><span class="line">    minSum = <span class="number">1e10</span></span><br><span class="line">    minS0 = <span class="number">0.0</span></span><br><span class="line">    minBeta = <span class="number">0.0</span></span><br><span class="line">    bestRes = <span class="literal">None</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span> S <span class="keyword">in</span> S0:</span><br><span class="line">        <span class="keyword">for</span> b <span class="keyword">in</span> beta:</span><br><span class="line">            <span class="comment"># 模型的差分方程</span></span><br><span class="line">            <span class="function"><span class="keyword">def</span> <span class="title">diff_eqs_2</span>(<span class="params">INP, t</span>):</span></span><br><span class="line">                Y = np.zeros((<span class="number">3</span>))</span><br><span class="line">                V = INP</span><br><span class="line">                Y[<span class="number">0</span>] = -b * V[<span class="number">0</span>] * V[<span class="number">1</span>]</span><br><span class="line">                Y[<span class="number">1</span>] = b * V[<span class="number">0</span>] * V[<span class="number">1</span>] - gamma * V[<span class="number">1</span>]</span><br><span class="line">                Y[<span class="number">2</span>] = gamma * V[<span class="number">1</span>]</span><br><span class="line">                <span class="keyword">return</span> Y</span><br><span class="line"></span><br><span class="line">            <span class="comment"># 数值解模型方程</span></span><br><span class="line">            INPUT = [S, I0, <span class="number">0.0</span>]</span><br><span class="line">            RES = spi.odeint(diff_eqs_2, INPUT, t_range)</span><br><span class="line">            errsum = error(RES[:<span class="number">21</span>, <span class="number">1</span>])</span><br><span class="line">            <span class="keyword">if</span> errsum &lt; minSum:</span><br><span class="line">                minSum = errsum</span><br><span class="line">                minS0 = S</span><br><span class="line">                minBeta = b</span><br><span class="line">                bestRes = RES</span><br><span class="line">                print(<span class="string">&quot;S0=%d beta=%f minErr=%f&quot;</span> % (S, b, errsum))</span><br><span class="line">    print(<span class="string">&quot;S0 = %d β = %f&quot;</span> % (minS0, minBeta))</span><br></pre></td></tr></table></figure>
<p>结果 S0 = 39000, β = 8e-6<br>上述程序耗时较长，只在探索时执行，完了就注释掉，用最优参数进行预测。<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/04.png"><br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/05.png"><br>预测最大感染人数:23769 时间是在1月10日的33天后，也就是2月12日。<br>本文代码:<a target="_blank" rel="noopener" href="https://github.com/zwdnet/2019-nCov-SIRmodel">https://github.com/zwdnet/2019-nCov-SIRmodel</a></p>
<p><strong>再次声明:本文只是我个人在家无聊的游戏作品，不是正儿八经的预测。我也不是流行病学专业人士。祝疫情早日结束！武汉加油！中国加油！</strong></p>
<p>2020年3月3日补充<br>一直在每天更新官方的疫情数据，顶峰到来的时间比预测的晚了十来天，顶峰人数也不对。尤其是有天暴增了一万多确诊的。不过趋势没变。模型预测趋势还是可以的。现在就希望别再从国外输入回来啦。<br><img src="https://zymblog-1258069789.cos.ap-chengdu.myqcloud.com/blog0188-SIRmodel/06.png"></p>
<p>参考文献:<br>[1]SIR模型实现, <a target="_blank" rel="noopener" href="https://blog.csdn.net/huozi07/article/details/50450433">https://blog.csdn.net/huozi07/article/details/50450433</a><br>[2]百度百科SIR模型词条, <a target="_blank" rel="noopener" href="https://baike.baidu.com/item/SIR%E6%A8%A1%E5%9E%8B">https://baike.baidu.com/item/SIR%E6%A8%A1%E5%9E%8B</a><br>[3]疫情通报.<a target="_blank" rel="noopener" href="http://www.nhc.gov.cn/xcs/yqtb/list_gzbd.shtml">http://www.nhc.gov.cn/xcs/yqtb/list_gzbd.shtml</a><br>[4]计算流行病学. <a target="_blank" rel="noopener" href="https://www.csdn.net/article/1970-01-01/2816565">https://www.csdn.net/article/1970-01-01/2816565</a><br>[5]关于新型冠状病毒（2019-nCoV）疫情的《国际卫 生条例（2005）》突发事件委员会会议的声明. <a target="_blank" rel="noopener" href="https://www.who.int/zh/news-room/detail/23-01-2020-statement-on-the-meeting-of-the-international-health-regulations-(2005)-emergency-committee-regarding-the-outbreak-of-novel-coronavirus-(2019-ncov)">https://www.who.int/zh/news-room/detail/23-01-2020-statement-on-the-meeting-of-the-international-health-regulations-(2005)-emergency-committee-regarding-the-outbreak-of-novel-coronavirus-(2019-ncov)</a><br>[6]Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia. <a target="_blank" rel="noopener" href="https://www.nejm.org/doi/full/10.1056/NEJMoa2001316?query=featured_home">https://www.nejm.org/doi/full/10.1056/NEJMoa2001316?query=featured_home</a><br>[7]基于SIR模型对新型冠状病毒疫情趋势的简单分析.<a target="_blank" rel="noopener" href="https://zhuanlan.zhihu.com/p/104379096">https://zhuanlan.zhihu.com/p/104379096</a></p>
<p>我发文章的四个地方，欢迎大家在朋友圈等地方分享，欢迎点“在看”。<br>我的个人博客地址：<a href="https://zwdnet.github.io/">https://zwdnet.github.io</a><br>我的知乎文章地址： <a target="_blank" rel="noopener" href="https://www.zhihu.com/people/zhao-you-min/posts">https://www.zhihu.com/people/zhao-you-min/posts</a><br>我的博客园博客地址： <a target="_blank" rel="noopener" href="https://www.cnblogs.com/zwdnet/">https://www.cnblogs.com/zwdnet/</a><br>我的微信个人订阅号：赵瑜敏的口腔医学学习园地</p>
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